* Air France-KLM reiterates strong interest in TAP

* Hands-off approach may be key in battle for TAP

* Portugal wants partner who will retain local identity

* Lufthansa, IAG seen as strong competitors

LONDON/LISBON, Oct 6 (Reuters) - Air France-KLM's decision to take a stake in Scandinavia's SAS airline offers a taste of its potential approach for the next big airline takeover battle in Europe: the fight to resuscitate Portuguese carrier TAP.

Portugal's government said last Thursday it plans to sell at least a 51% stake in state-owned TAP, after the cabinet approved the legal framework for the privatisation process.

Europe's national carriers have struggled to compete with low-cost airlines like Ryanair and Wizz Air and larger groups like Lufthansa, Air France-KLM and IAG have come in to revamp and save them.

Tuesday's SAS deal, which saw U.S. investment firm Castlelake and Air France-KLM enter as new major shareholders alongside the Danish state, was another long-awaited shake up for one of Europe's legacy brands.

Air France-KLM is only taking a 19.9% stake in SAS and may have limited influence in revamping the airline, which has struggled with fragmentation across its Danish and Swedish hubs.

Still, this arms-length approach may prove appealing to TAP.

Air France-KLM has tended to let airlines it invests in keep their operations and branding. Following the 2004 merger between the French and Dutch carriers, both still fly under their own livery and many operations remain separate.

Germany's Lufthansa and Anglo Spanish IAG, other likely competitors in the battle for TAP, have been known for deeper restructurings of the airlines they takeover, streamlining business practices and branding.

"It will depend on what promises Air France-KLM make on keeping the brand alive, keeping the brand separate and keeping the operations in Portugal and connections in Portugal," said James Halstead, an aviation analyst.

Analysts say the SAS deal is a success for Air France-KLM as they have peeled one airline away from Lufthansa's sphere of influence in Northern Europe and from airline grouping Star Alliance.

Now, they're trying to repeat their success, poaching another Star Alliance member away from Lufthansa with TAP but facing a higher price and fierce competition.

Air France-KLM reiterated its strong interest in TAP on Thursday, adding the purchase of a stake in SAS did not affect its ability to take part in TAP's privatisation.

Portugal's government is keen to find a partner that will allow it to save its struggling airline without losing its branding, local hubs and national Portuguese identity.

A more hands-off approach could be the golden ticket to tapping into TAP's lucrative south American connections, analysts and investors say.

"TAP's priority is the 'integrity' of TAP, which would probably have its best chance of success in an Air France-KLM partnership," an investor source said.

With the process still in its infancy, Air France-KLM has a chance to make a pitch to TAP now off the back of SAS.

"Everything is still at a very, very early stage," a source with knowledge of the matter told Reuters.

"The effective work to understand which airlines are truly interested, the real market appetite, and the strategies of the different players has not yet begun."

LONG FIGHT AHEAD

There are remaining challenges for Air France-KLM as it looks to clear regulatory hurdles with SAS, including approval from American and European policymakers. A hiccup could drag down the process for the TAP talks, too.

U.S. private equity giant Apollo Global Management, which last year provided SAS with $700 million in bridge financing and had bid to take a majority stake in the airline, faced European Union rules preventing non-EU investors from taking control of an airline.

Europe's airline industry has a history of cultural and political barriers to smooth integration, even within Air France-KLM itself. How it manages to negotiate across Swedish and Danish cultures with less than a 20% stake remains to be seen.

The other two bidders in play also have strong advantages for TAP.

"One thing is certain: Portugal would want the buyer of TAP to be a reputable and strong airline...and at the same time enhance not only the Lisbon hub, but also other airports like Porto," the source with knowledge of the Portuguese government's position said.

IAG could appeal with its strong presence in the south Atlantic, with Iberia holding a strong connection to Spanish-speaking parts of South America.

Lufthansa's membership in global airline network Star Alliance, of which TAP is also a part, could also make it an attractive partner and could ease the transition for the Portuguese airline.

Air France-KLM is a member of the rival SkyTeam alliance.

"TAP does not have a future alone," the source said. "TAP must be part of an alliance like this." (Additional reporting by Jacob Gronholt-Pedersen and Tim Hepher; Editing by Matt Scuffham and Sharon Singleton)